I keep wondering whether Sage is making a mistake, or I'm not understanding complex analysis. I'm a little afraid to learn the answer. :)
Take f(z) = (z-I)*(z-1)^2/(z-(-1/2-I/3)). It's analytic everywhere
except at -1/2-I/3, where it has a simple pole. So, if I integrate over
a circle centered at 0 of radius, say, 2, the answer should be
2*pi*I*(residue of f at -1/2 - I/3),
which is pi*(181/27 + 19*I/36). However, when I try to do the contour
integral, I get:
sage: integrate(f(2*exp(I*t)) * 2*I*exp(I*t), (t, 0, 2*pi))
0
even though the contour encloses the pole. It works if I center the
circle around the pole:
sage: integrate(f(-1/2-I/3 + exp(I*t)) * I*exp(I*t), (t, 0, 2*pi))
(19/36*I + 181/27)*pi
and also if I integrate over the square with vertices 1+i, 1-i, -1-i,
-1+i. What's wrong with the circle at the origin?
Note that Mathematica gets this right, although you need to ask for full
simplification: with f[z_] = (z-I)*(z-1)^2/(z-(-1/2-I/3)), you get
In[5]:= Integrate[f[2*Exp[I*t]] * 2*I*Exp[I*t], {t, 0, 2*Pi}]//FullSimplify
181 19 I
Out[5]= (--- + ----) Pi
27 36
Any ideas?
Dan
--
--- Dan Drake
----- http://mathsci.kaist.ac.kr/~drake
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